216 SECRETS OF EARTH AND SEA 



" streptocones," and their S-like boundary " a hemicyclic 

 sigmoid." As shown in Fig. 56, by drawing a second 

 hemicyclic sigmoid of the same dimensions at right angles 

 to the first, the circle is divided into four smaller strepto- 

 cones. By using sigmoids or half-sigmoids of a curvature 

 of a different order from that of the hemicyclic one, but of 

 a precisely defined nature, the circle may be divided into 

 three, six, eight or more equal " streptocones " of graceful 

 proportions, some of which have been used either in series 

 as borders in metal work (for circular dishes and goblets) or 

 as detached or grouped elements in pattern-designs (stone- 

 work tracery, embroidery, woven and printed fabrics). 



Apart from this development of the " streptocone " as 

 an important feature in decorative work, it is not without 

 interest in connection with the probable importance and 

 significance of the Japanese double streptocone, as we may 

 call the Tomoye, to note some of its geometrical features. 

 Referring to the Fig. 60, it is obvious that each of the 

 paired streptocones is equal in area to half the enclosing 

 circle, also that each of the two inscribed circles (a, b) has 

 an area of one-fourth of that of the enclosing circle and 

 that each arbelus (c, d} has also an area one-fourth that 

 of the enclosing circle and is equal in area to each of the 

 inscribed circles (a, b). Each of the two constituent 

 " streptocones " is made up of a complete circle capped by 

 an " arbelus " equal in area to it (namely, one-quarter of 

 that of the big circle). It is obvious that the area of the 

 arbelus formed in a semicircle by two enclosed semi- 

 circles which are contiguous and of equal base as in Fig. 60, 

 is equal to that of a circle the diameter of which is the 

 vertical line drawn from the apex of the arbelus to the arc 

 of the semicircle (Fig 60). This is true whether the 

 enclosed contiguous semicircles have chords of equal or 

 unequal length (Fig. 60). This fact was known to the 

 Greek geometricians, as I am informed by Sir Thos. Heath. 



